Chapter 16 QUANTUM MECHANICS

By exercising these rights, you accept By Matthew Raspanti Chapter 16 and agree to be bound by the terms Attribution-NonCommercial-ShareAlike and conditions of this Creative Commons Attribution- NonCommercial-ShareAlike 4.0 QUANTUM MECHANICS International Public License. To the extent this Public License may be interpreted as a contract, You are granted the Licensed Rights in consideration of Your acceptance of these terms and conditions, and the Licensor grants You such rights in consideration of benefits the Licensor receives from making the Licensed Material available under these terms and conditions. "Those who are not shocked when they first come across quantum theory cannot possibly have understood it." Niels Bohr "I think I can safely say that nobody understands quantum mechanics." Richard Feynman Empirical Refinements of Bohr's model Bohr's achievement in 1913 was remarkable, but his theory was by no means complete. It did not account, for instance, for the fact that some spectral lines are brighter than others; nor could it be applied to more complex atoms. Bohr tried for many years to develop a mathematical theory for helium, which has the second simplest atom (a nucleus and two electrons). When more than one electron is involved, the situation becomes much more complicated because, in addition to the attractive forces between the positive nucleus and the electrons, one must consider the repulsive forces among the negative electrons. In the years that followed Bohr's discovery, physicists extended his theory, in order to account for various features of atomic spectra. Defining the "state" of an electron became more complicated than specifying one of a number of concentric circular orbits, all on the same plane. Whereas a single "quantum number" had sufficed for Bohr's theory, four quantum numbers, eventually, had to be used. Three were associated with the size, shape and orientation of the allowed electronic orbits, which could be circular or elliptical. The fourth quantum number was associated with a spinning motion of the electron about its axis. Quantum numbers were assigned to the electron using rules that were discovered empirically. What was lacking was a sound mathematical foundation. Einstein's Statistics of Electron Excitation In 1916, after he completed his General Theory of Relativity, Einstein turned his Matthew Raspanti Physics for Beginners 112 attention again to quantum theory. He used statistical techniques and Bohr's model of the atom to study the behavior of huge numbers of atoms. According to Bohr, in a hot gas, as countless atoms constantly collide with one another, electrons are excited to higher energy levels; later, they fall back to lower levels and emit radiation (photons). Einstein's statistical approach could explain why some spectral lines are brighter than others, the reason being that some transitions between energy states are more likely to happen than others. After Maxwell, reality seemed to consist of empty space populated by two totally different kinds of "things": particles and waves. Particles were thought of as being point-like and having such properties as energy, momentum, mass and, possibly, an electrical charge. Waves, on the other hand, were thought of as being spread out in space, and having such properties as energy, amplitude, frequency, wavelength and speed of propagation. In 1905, Einstein upset this sharp distinction. As we saw, he proposed that light - whose wave-like nature had been accepted after Young's 2-slit experiment - could be emitted or absorbed only as discrete quanta of energy. In 1909, he started talking of "point-like" quanta or massless particles, later called photons. Einstein's statistical calculations of how matter absorbs or emits radiation explained how momentum could be transferred from a photon to an electron. It was necessary, however, to assume that each photon carried with it a momentum which was proportional to frequency. Momentum had been considered before to be a particle-like property, normally expressed as mass x velocity. The massless photon was seen now as something that had both a particle-like property (momentum) and a wave-like property (frequency). Prince de Broglie In 1924, while still a graduate student in physics, a French nobleman, Louis de Broglie (pronounced de Broy), proposed a brilliantly bold conjecture. If light waves can behave like particles, why shouldn't particles, such as electrons, behave like waves? Why couldn't an electron be a wave? Prince de Broglie was born in 1892 in one of Europe's most aristocratic families. For centuries, his family had contributed diplomats, cabinet ministers and generals. After he entered the University of Paris in 1910, inspired by his brother, who was a physicist, he developed an interest in physics and in Einstein's work. His idea of a wave/particle duality became the basis of the doctoral thesis he submitted in 1924. De Broglie proposed that electrons as well as photons have associated Matthew Raspanti Chapter 16: Quantum Mechanics 113 with them both a momentum and a wavelength6. He could not say what was the physical significance of an electron's wavelength, but he could show an interesting connection between it and Bohr's orbits. The circular orbits that were allowed in Bohr's theory turned out to be those that could contain exactly a whole number of wavelengths. Even for a doctoral thesis, de Broglie's proposal was too original to be comfortably accepted by the examiners without any experimental support. A copy of the thesis was sent to Einstein, who commented "It may look crazy, but it really is sound". (Years later, shortly before his death, commenting about some controversial new theory, Bohr remarked that the theory was certainly crazy, but wondered whether it was crazy enough to have merit.) In 1927, confirmation of de Broglie's electron wave was provided by experiments performed by Clinton Davisson at Bell Telephone Laboratories in New York. In 1929, de Broglie became the first to receive a Nobel Prize for a doctoral thesis. In 1937, Davisson shared the Nobel Prize with George Thomson, who independently had found confirmation for the electron wave. Interestingly, in 1906, J.J. Thomson had received the Nobel Prize for proving that electrons are particles; 31 years later, his son received the Nobel Prize for proving that electrons are waves! The reader may be totally mystified by the idea of a "particle" behaving like a "wave", but so was de Broglie who proposed the idea, and so were the physicists of his day. QUANTUM MECHANICS At the beginning of 1925, the quantum theory that started with Bohr in 1913 was still a hodgepodge of hypotheses and cookbook rules. In the 12 months following June 1925, quantum theory finally gained a firm mathematical footing. Not one but two theories emerged, independently of one another. Although their approaches seemed at first sharply different, the two theories were later proved to be equivalent aspects of a single theory. Now called "quantum mechanics", this combined theory was the creation of a new generation of physicists, mostly born since Planck's discovery of the quantum. Their youth made them scientific revolutionaries willing to break away from classical physics. Bohr was the undisputed guiding spirit of this extraordinary development, and Copenhagen became its center. The Gottingen Trio The first of the two mathematical theories was developed by three German 6 For light or, more generally, electromagnetic waves, we can talk in terms of either frequency or wavelength because the product wavelength x frequency is a constant, the speed of light c. Matthew Raspanti Physics for Beginners 114 physicists at the University of Gottingen: Werner Heisenberg (1901-1976), Max Born (1882-1970), and Pascual Jordan (1902-1980). Heisenberg, who was at the time a research assistant to Max Born, proposed a radical reinterpretation of the basic concepts of mechanics with regard to atomic particles. The new approach was guided by the principle that a physical theory is obligated to consider only those things that can actually be observed. There is no way of observing directly an electron orbiting around the nucleus of an atom. What can be observed are spectral lines, which are interpreted in terms of electrons jumping from one energy level to another. The notion of tiny balls in orbits is a convenient mental image that is superimposed on the actual observations, because that is how we see things moving in our everyday world. In developing his theory, Heisenberg was willing to abandon convenient analogies and mental images, and to follow a totally abstract approach. The breakthrough came in June of 1925 while he was recovering from a severe attack of hay fever on a North Sea island. There, away from distractions, he was able to concentrate on the new ideas that were forming in his mind. Within three months, the collaboration of Heisenberg, Born and Jordan resulted in the publication of a comprehensive paper. Schrodinger's Equation Only months later, Erwin Schrodinger (Austrian, 1887-1961) published a different mathematical theory he had developed independently of the Gottingen group. He had been inspired by Einstein's support of de Broglie's theory. Unlike the Gottingen group, he was guided by mental images of atoms with electrons as waves. Before long, the two theories were proved to be fully equivalent. Being much simpler to use, Schrodinger's equation became the preferred mathematical tool of quantum mechanics to solve atomic problems. The equation gave a very good account of the spectrum of the hydrogen atom in a way that was mathematically consistent. But even for the very simple case of hydrogen, the mathematics involved is very difficult. To convey just a flavor of this complex theory, it will suffice to summarize the results as follows. Each allowed "state" of the electron is some configuration identified by a particular set of values for three "quantum numbers" n, l (the letter l) and m.

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